本文是https://towardsdatascience.com/how-to-do-deep-learning-on-graphs-with-graph-convolutional-networks-7d2250723780第一部分A High-Level Introduction to Graph Convolutional Networks的翻譯和總結，介紹了簡單的Graph Convolutional Network，並以空手道俱樂部數據集爲示例進行實現分析，部分添加修改。
What is a Graph Convolutional Network?
可以簡單表示爲
A Simple Graph Example
以下圖爲例
A爲該圖的鄰接矩陣
import numpy as np
A = np.matrix([ ##鄰接矩陣
[0, 1, 0, 0],
[0, 0, 1, 1],
[0, 1, 0, 0],
[1, 0, 1, 0]],
dtype=float
)
X爲輸入的特徵向量，我們直接取值，維度爲,其中N爲結點個數，爲輸入向量的特徵維數
X = np.matrix([
[i, -i]
for i in range(A.shape[0])
], dtype=float)
X
matrix([[ 0., 0.],
[ 1., -1.],
[ 2., -2.],
[ 3., -3.]])
A*X
matrix([[ 1., -1.],
[ 5., -5.],
[ 1., -1.],
[ 2., -2.]])
我們可以發現，每行，就是每個結點的特徵是它周圍結點的特徵和
比如，結點1和結點2，3鄰接，所以A*X第2行 5 = [0, 0, 1, 1] *[0 ,1 ,2 ,3]T
但是有兩點問題
- A*X的結點表示中，並沒有加自己的特徵值。只有有self-loop 的結構纔會把自己特徵值包含在內
- 鄰接結點多的結點的特徵值會大，少的特徵值就小
第一個問題可以通過加Self-Loops來解決
I = np.matrix(np.eye(A.shape[0]))
I
matrix([[1., 0., 0., 0.],
[0., 1., 0., 0.],
[0., 0., 1., 0.],
[0., 0., 0., 1.]])
A_hat = A + I
A_hat * X
matrix([[ 1., -1.],
[ 6., -6.],
[ 3., -3.],
[ 5., -5.]])
第二個問題可以通過歸一化特徵表示來解決
原因見https://tkipf.github.io/graph-convolutional-networks/
D = np.array(np.sum(A, axis=0))[0]
D = np.matrix(np.diag(D))
D
matrix([[1., 0., 0., 0.],
[0., 2., 0., 0.],
[0., 0., 2., 0.],
[0., 0., 0., 1.]])
歸一化前
A
matrix([[0., 1., 0., 0.],
[0., 0., 1., 1.],
[0., 1., 0., 0.],
[1., 0., 1., 0.]])
歸一化後
D**-1 * A
matrix([[0. , 1. , 0. , 0. ],
[0. , 0. , 0.5, 0.5],
[0. , 0.5, 0. , 0. ],
[1. , 0. , 1. , 0. ]])
D**-1 * A * X
matrix([[ 1. , -1. ],
[ 2.5, -2.5],
[ 0.5, -0.5],
[ 2. , -2. ]])
Putting it All Together
接下來我們把兩個改進方案都運用起來
A_hat = A + I
D_hat = np.array(np.sum(A_hat, axis=0))[0]
D_hat = np.matrix(np.diag(D_hat))
D_hat
matrix([[2., 0., 0., 0.],
[0., 3., 0., 0.],
[0., 0., 3., 0.],
[0., 0., 0., 2.]])
W = np.matrix([
[1, -1],
[-1, 1]
])
X
matrix([[ 0., 0.],
[ 1., -1.],
[ 2., -2.],
[ 3., -3.]])
D_hat**-1 * A_hat*X*W
matrix([[ 1., -1.],
[ 4., -4.],
[ 2., -2.],
[ 5., -5.]])
Back to Reality
現在我們在現實中的網絡中運用圖卷集網絡(graph convolutional network技術。選取的網絡爲空手道俱樂部數據集(karate_club_graph)。
from networkx import to_numpy_matrix
import networkx as nx
zkc = nx.karate_club_graph()
order = sorted(list(zkc.nodes()))
A = to_numpy_matrix(zkc, nodelist=order)
I = np.eye(zkc.number_of_nodes())
A_hat = A + I
D_hat = np.array(np.sum(A_hat, axis=0))[0]
D_hat = np.matrix(np.diag(D_hat))
def plot_graph(G, weight_name=None):
'''
G: a networkx G
weight_name: name of the attribute for plotting edge weights (if G is weighted)
'''
%matplotlib notebook
import matplotlib.pyplot as plt
plt.figure()
pos = nx.spring_layout(G)
edges = G.edges()
weights = None
if weight_name:
weights = [int(G[u][v][weight_name]) for u,v in edges]
labels = nx.get_edge_attributes(G,weight_name)
nx.draw_networkx_edge_labels(G,pos,edge_labels=labels)
nx.draw_networkx(G, pos, edges=edges, width=weights);
else:
nodelist1 = []
nodelist2 = []
for i in range (34):
if zkc.nodes[i]['club'] == 'Mr. Hi':
nodelist1.append(i)
else:
nodelist2.append(i)
#nx.draw_networkx(G, pos, edges=edges);
nx.draw_networkx_nodes(G, pos, nodelist=nodelist1, node_size=300, node_color='r',alpha = 0.8)
nx.draw_networkx_nodes(G, pos, nodelist=nodelist2, node_size=300, node_color='b',alpha = 0.8)
nx.draw_networkx_edges(G, pos, edgelist=edges,alpha =0.4)
plot_graph(zkc)
<IPython.core.display.Javascript object>
W_1 = np.random.normal(
loc=0, scale=1, size=(zkc.number_of_nodes(), 4))
W_2 = np.random.normal(
loc=0, size=(W_1.shape[1], 2))
def relu(x):
return (abs(x) + x) / 2
def gcn_layer(A_hat, D_hat, X, W):
return relu(D_hat**-1 * A_hat * X * W)
H_1 = gcn_layer(A_hat, D_hat, I, W_1)
H_2 = gcn_layer(A_hat, D_hat, H_1, W_2)
output = H_2
feature_representations = {
node: np.array(output)[node]
for node in zkc.nodes()}
feature_representations
{0: array([0.88602091, 0.34237275]),
1: array([0.40862582, 0. ]),
2: array([0.38693926, 0. ]),
3: array([0.19478099, 0.10516756]),
4: array([0.82815959, 0.41738152]),
5: array([1.1971192 , 0.46978126]),
6: array([1.2271154 , 0.63378424]),
7: array([0., 0.]),
8: array([0.11110005, 0. ]),
9: array([0., 0.]),
10: array([0.6209274 , 0.26495055]),
11: array([1.60869786, 0.79829349]),
12: array([0.35029305, 0.56226336]),
13: array([0.02171053, 0. ]),
14: array([0. , 0.02638456]),
15: array([0.06979159, 0.68002892]),
16: array([1.7675629 , 0.82039984]),
17: array([0.50286326, 0. ]),
18: array([0.31509428, 0.29327311]),
19: array([0.37260057, 0. ]),
20: array([0., 0.]),
21: array([0.70826438, 0.10767323]),
22: array([0.15022781, 0.25590783]),
23: array([0.17645064, 0.16650816]),
24: array([0.29110197, 0.20382017]),
25: array([0.18688296, 0.14564473]),
26: array([0.02367803, 0.17550985]),
27: array([0., 0.]),
28: array([0.51547931, 0. ]),
29: array([0.05318727, 0.16647217]),
30: array([0.31639705, 0. ]),
31: array([0.24761528, 0.03619812]),
32: array([0.48872535, 0.31039692]),
33: array([0.62804696, 0.26496685])}
import matplotlib.pyplot as plt
%matplotlib notebook
for i in range (34):
if zkc.nodes[i]['club'] == 'Mr. Hi':
plt.scatter(np.array(output)[i,0],np.array(output)[i,1] ,color = 'b',alpha=0.5,s = 100)
else:
plt.scatter(np.array(output)[i,0],np.array(output)[i,1] ,color = 'r',alpha=0.5,s = 100)
#plt.scatter(np.array(output)[:,0],np.array(output)[:,1])
<IPython.core.display.Javascript object>
目前來看，這個映射分類效果並不好，待我後續分析補充吧。